
vergangen
past
Refers to something that lies in the past, has happened, or no longer exists; previous.
In den vergangenen Wochen hat es kaum geregnet.
In the past few weeks, it has hardly rained.
past
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past
Refers to something that lies in the past, has happened, or no longer exists; previous.
In den vergangenen Wochen hat es kaum geregnet.
In the past few weeks, it has hardly rained.
'Vergangen' (past) is the past participle of the verb 'vergehen', which means 'to pass' (for time). It is used like an adjective to express that something belongs to the past.
Die Zeit vergeht so schnell, schon wieder ist ein Jahr vergangen.
Time passes so quickly; another year has already passed.
The adjective 'vergangen' (past) is almost always combined with nouns that describe a period of time. Typical combinations are 'vergangene Woche' (last week), 'vergangener Monat' (last month), or 'vergangenes Jahr' (last year).
Am vergangenen Wochenende war ich im Museum.
Last weekend I was at the museum.
'Vergangen' (past) is mainly used attributively, meaning it stands before a noun and is declined like a normal adjective. Predicative use ('Die Zeit ist vergangen' — time has passed) is usually part of the perfect tense of the verb 'vergehen' (to pass).
Die Ereignisse der vergangenen Tage haben alle überrascht.
The events of the past few days surprised everyone.
last
Both words can refer to a period of time that has passed. „Letzt“ often refers to the most recent period and, depending on the context, may also suggest a conclusion, whereas „vergangen“ more generally only says that the period is over. Use „letzt“ for temporal proximity and „vergangen“ for a neutral look back.

Im vergangenen Monat haben wir besonders viel gearbeitet.
We worked particularly hard in the past month.
Im letzten Monat haben wir besonders viel gearbeitet.
We worked particularly hard last month.